Explicit partial and functional differential equations for beables or observables

Abstract

We provide explicit partial differential equations - in finite cases - and functional differential equations - in field-theoretic cases - which determine observables or beables in the senses of Kuchař and of Dirac. These cover a wide range of relational mechanics models as well as Electromagnetism, Yang--Mills Theory and General Relativity. We give an underlying reason why pure-configuration Kuchař observables are already well-known: various types of shape, E-fields, B-fields, loops and 3-geometries. The partial differential equations or functional differential equations for pure-momentum observables are also posed, as are those for observables which have a mixture of configuration and momentum functional dependence.

9 pages including 4 figures: minor presentational upgrades and further references added

References in corpus (5)

• Problem of Time and Background Independence: the Individual Facets
• Configuration Spaces in Fundamental Physics
• TRiPoD (Temporal Relationalism incorporating Principles of Dynamics)
• Origin of Structure in the Universe: Quantum Cosmology Reconsidered
• Where to Apply Relationalism

Cited by in corpus (19)

• A Local Resolution of the Problem of Time
• The Smallest Shape Spaces. I. Shape Theory Posed, with Example of 3 Points on the Line
• On Types of Observables in Constrained Theories
• The Smallest Shape Spaces. III. Triangles in 2- and 3-d
• A Local Resolution of the Problem of Time. III. The other classical facets piecemeal
• Specific PDEs for Preserved Quantities in Geometry. I. Similarities and Subgroups
• A Local Resolution of the Problem of Time. I. Introduction and Temporal Relationalism
• Spaces of Observables from Solving PDEs. I. Translation-Invariant Theory
• Geometry from Brackets Consistency
• Monopoles of Twelve Types in 3-Body Problems
• $N$-Body Problem: Minimal $N$'s for Qualitative Nontrivialities
• Specific PDEs for Preserved Quantities in Geometry. II. Affine Transformations and Subgroups
• A Local Resolution of the Problem of Time. XIV. Grounding on Lie's Mathematics
• Specific PDEs for Preserved Quantities in Geometry. III. 1-d Projective Transformations and Subgroups
• Problem of Time: Temporal Relationalism Compatibility of Other Local Classical Facets Completed
• Topological Shape Theory
• Rubber Relationalism: Smallest Graph-Theoretically Nontrivial Leibniz Spaces
• Quadrilaterals in Shape Theory. II. Alternative Derivations of Shape Space: Successes and Limitations
• A Local Resolution of the Problem of Time. VIII. Assignment of Observables